nag_tsa_varma_diagnostic (g13dsc) is a diagnostic checking function suitable for use after fitting a vector ARMA model to a multivariate time series using nag_tsa_varma_estimate (g13ddc). The residual cross-correlation matrices are returned along with an estimate of their asymptotic standard errors and correlations. Also, nag_tsa_varma_diagnostic (g13dsc) calculates the modified Li–McLeod portmanteau statistic and its significance level for testing model adequacy.

2

Specification

#include <nag.h>

#include <nagg13.h>

void

nag_tsa_varma_diagnostic (Integer k, Integer n, const double v[], Integer kmax, Integer ip, Integer iq, Integer m, const double par[], const Nag_Boolean parhld[], double qq[], Integer ishow, const char *outfile, double r0[], double r[], double rcm[], Integer pdrcm, double *chi, Integer *idf, double *siglev, NagError *fail)

3

Description

Let

W_{t} = {(w_{1 t}, w_{2 t}, \dots, w_{k t})}^{T}

, for

t = 1, 2, \dots, n

, denote a vector of

k

time series which is assumed to follow a multivariate ARMA model of the form

\begin{array}{l} W_{t} - μ = & ϕ_{1} (W_{t - 1} - μ) + ϕ_{2} (W_{t - 2} - μ) + \dots + ϕ_{p} (W_{t - p} - μ) \\ + ε_{t} - θ_{1} ε_{t - 1} - θ_{2} ε_{t - 2} - \dots - θ_{q} ε_{t - q}, \end{array}

(1)

where

ε_{t} = {(ε_{1 t}, ε_{2 t}, \dots, ε_{k t})}^{T}

, for

t = 1, 2, \dots, n

, is a vector of

k

residual series assumed to be Normally distributed with zero mean and positive definite covariance matrix

Σ

. The components of

ε_{t}

are assumed to be uncorrelated at non-simultaneous lags. The

ϕ_{i}

and

θ_{j}

are

k

k

matrices of parameters.

\{ϕ_{i}\}

, for

i = 1, 2, \dots, p

, are called the autoregressive (AR) parameter matrices, and

\{θ_{i}\}

, for

i = 1, 2, \dots, q

, the moving average (MA) parameter matrices. The parameters in the model are thus the

p

(

k

k

)

ϕ

-matrices, the

q

(

k

k

)

θ

-matrices, the mean vector

μ

and the residual error covariance matrix

Σ

. Let

A (ϕ) = {[\begin{array}{l} ϕ_{1} & I & 0 & . & . & . & 0 \\ ϕ_{2} & 0 & I & 0 & . & . & 0 \\ . & . \\ . & . \\ . & . \\ ϕ_{p - 1} & 0 & . & . & . & 0 & I \\ ϕ_{p} & 0 & . & . & . & 0 & 0 \end{array}]}_{p k \times p k} and B (θ) = {[\begin{array}{l} θ_{1} & I & 0 & . & . & . & 0 \\ θ_{2} & 0 & I & 0 & . & . & 0 \\ . & . \\ . & . \\ . & . \\ θ_{q - 1} & 0 & . & . & . & I \\ θ_{q} & 0 & . & . & . & . & 0 \end{array}]}_{q k \times q k}

where

I

denotes the

k

k

identity matrix.

The ARMA model (1) is said to be stationary if the eigenvalues of

A (ϕ)

lie inside the unit circle, and invertible if the eigenvalues of

B (θ)

lie inside the unit circle. The ARMA model is assumed to be both stationary and invertible. Note that some of the elements of the

ϕ

- and/or

θ

-matrices may have been fixed at pre-specified values (for example by calling nag_tsa_varma_estimate (g13ddc)).

The estimated residual cross-correlation matrix at lag

l

is defined to the

k

k

matrix

{\hat{R}}_{l}

whose

(i, j)

th element is computed as

{\hat{r}}_{i j} (l) = \frac{\sum_{t = l + 1}^{n} ({\hat{ε}}_{i t - l} - {\bar{ε}}_{i}) ({\hat{ε}}_{j t} - {\bar{ε}}_{j})}{\sqrt{\sum_{t = 1}^{n} {({\hat{ε}}_{i t} - {\bar{ε}}_{i})}^{2} \sum_{t = 1}^{n} {({\hat{ε}}_{j t} - {\bar{ε}}_{j})}^{2}}}, l = 0, 1, \dots, i ​ and ​ j = 1, 2, \dots, k,

where

{\hat{ε}}_{i t}

denotes an estimate of the

t

th residual for the

i

th series

ε_{i t}

and

{\bar{ε}}_{i} = \sum_{t = 1}^{n} {\hat{ε}}_{i t} / n

. (Note that

{\hat{R}}_{l}

is an estimate of

E (ε_{t - l} ε_{t}^{T})

, where

E

is the expected value.)

A modified portmanteau statistic,

Q_{(m)}^{*}

, is calculated from the formula (see Li and McLeod (1981))

Q_{(m)}^{*} = \frac{k^{2} m (m + 1)}{2 n} + n \sum_{l = 1}^{m} \hat{r} {(l)}^{T} ({\hat{R}}_{0}^{- 1} \otimes {\hat{R}}_{0}^{- 1}) \hat{r} (l),

where

\otimes

denotes Kronecker product,

{\hat{R}}_{0}

is the estimated residual cross-correlation matrix at lag zero and

\hat{r} (l) = vec ({\hat{R}}_{l}^{T})

, where

vec

of a

k

k

matrix is a vector with the

(i, j)

th element in position

(i - 1) k + j

m

denotes the number of residual cross-correlation matrices computed. (Advice on the choice of

m

is given in Section 9.2.) Let

l_{C}

denote the total number of ‘free’ parameters in the ARMA model excluding the mean,

μ

, and the residual error covariance matrix

Σ

. Then, under the hypothesis of model adequacy,

Q_{(m)}^{*}

, has an asymptotic

χ^{2}

-distribution on

m k^{2} - l_{C}

degrees of freedom.

Let

\underset{̲}{\hat{r}} = (vec (R_{1}^{T}), vec (R_{2}^{T}), \dots, vec (R_{m}^{T}))

then the covariance matrix of

\underset{̲}{\hat{r}}

is given by

Var (\hat{\underset{̲}{r}}) = [Y - X {(X^{T} G G^{T} X)}^{- 1} X^{T}] / n,

where

Y = I_{m} \otimes (Δ \otimes Δ)

and

G = I_{m} (G G^{T})

Δ

is the dispersion matrix

Σ

in correlation form and

G

a nonsingular

k

k

matrix such that

G G^{T} = Δ^{- 1}

and

G Δ G^{T} = I_{k}

. The construction of the matrix

X

is discussed in Li and McLeod (1981). (Note that the mean,

μ

, plays no part in calculating

Var (\hat{\underset{̲}{r}})

and therefore is not required as input to nag_tsa_varma_diagnostic (g13dsc).)

4

References

Li W K and McLeod A I (1981) Distribution of the residual autocorrelations in multivariate ARMA time series models J. Roy. Statist. Soc. Ser. B 43 231–239

5

Arguments

The output quantities k, n, v, kmax, ip, iq, par, parhld and qq from nag_tsa_varma_estimate (g13ddc) are suitable for input to nag_tsa_varma_diagnostic (g13dsc).

1: $k$ – IntegerInput

On entry:

k

, the number of residual time series.

Constraint:

k \geq 1

2: $n$ – IntegerInput

On entry:

n

, the number of observations in each residual series.

3: $v [kmax \times n]$ – const doubleInput

On entry:

v [kmax \times (t - 1) + i - 1]

must contain an estimate of the

i

th component of

ε_{t}

, for

i = 1, 2, \dots, k

and

t = 1, 2, \dots, n

Constraints:

no two rows of $v$ may be identical;
in each row there must be at least two distinct elements.

4: $kmax$ – IntegerInput

On entry: the first dimension of the arrays v, qq, r and r0 and the second dimension of the matrix r.

Constraint:

kmax \geq k

5: $ip$ – IntegerInput

On entry:

p

, the number of AR parameter matrices.

Constraint:

ip \geq 0

6: $iq$ – IntegerInput

On entry:

q

, the number of MA parameter matrices.

Constraint:

iq \geq 0

Note:

ip = iq = 0

is not permitted.

7: $m$ – IntegerInput

On entry: the value of

m

, the number of residual cross-correlation matrices to be computed. See Section 9.2 for advice on the choice of m.

Constraint:

ip + iq < m < n

8: $par [(ip + iq) \times k \times k]$ – const doubleInput

On entry: the parameter estimates read in row by row in the order

ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}

θ_{1}, θ_{2}, \dots, θ_{q}

Thus,

if $ip > 0$ , $par [(l - 1) \times k \times k + (i - 1) \times k + j - 1]$ must be set equal to an estimate of the $(i, j)$ th element of $ϕ_{l}$ , for $l = 1, 2, \dots, p$ and $i = 1, 2, \dots, k$ ;
if $iq \geq 0$ , $par [p \times k \times k + (l - 1) \times k \times k + (i - 1) \times k + j - 1]$ must be set equal to an estimate of the $(i, j)$ th element of $θ_{l}$ , for $l = 1, 2, \dots, q$ and $i = 1, 2, \dots, k$ .

The first

p \times k \times k

elements of par must satisfy the stationarity condition and the next

q \times k \times k

elements of par must satisfy the invertibility condition.

9: $parhld [(ip + iq) \times k \times k]$ – const Nag_BooleanInput

On entry:

parhld [i - 1]

must be set to Nag_TRUE if

par [i - 1]

has been held constant at a pre-specified value and Nag_FALSE if

par [i - 1]

is a free parameter, for

i = 1, 2, \dots, (p + q) \times k \times k

10: $qq [kmax \times k]$ – doubleInput/Output

On entry:

qq [kmax \times (j - 1) + i - 1]

is an efficient estimate of the

(i, j)

th element of

Σ

. The lower triangle only is needed.

Constraint:

qq

must be positive definite.

On exit: if

fail . code =

NE_G13D_AR, NE_G13D_DIAG, NE_G13D_FACT, NE_G13D_ITER, NE_G13D_MA, NE_G13D_RES, NE_G13D_ZERO_VAR or NE_NOT_POS_DEF, then the upper triangle is set equal to the lower triangle.

11: $ishow$ – IntegerInput

On entry: must be nonzero if the residual cross-correlation matrices

\{{\hat{r}}_{i j} (l)\}

and their standard errors

\{se ({\hat{r}}_{i j} (l))\}

, the modified portmanteau statistic with its significance and a summary table are to be printed. The summary table indicates which elements of the residual correlation matrices are significant at the

5 %

level in either a positive or negative direction; i.e., if

{\hat{r}}_{i j} (l) > 1.96 \times se ({\hat{r}}_{i j} (l))

then a ‘

+

’ is printed, if

{\hat{r}}_{i j} (l) < - 1.96 \times se ({\hat{r}}_{i j} (l))

then a ‘

-

’ is printed, otherwise a fullstop (.) is printed. The summary table is only printed if

k \leq 6

on entry.

The residual cross-correlation matrices, their standard errors and the modified portmanteau statistic with its significance are available also as output variables in r, rcm, chi, idf and siglev.

12: $outfile$ – const char *Input

On entry: the name of a file to which diagnostic output will be directed. If outfile is NULL the diagnostic output will be directed to standard output.

13: $r0 [kmax \times k]$ – doubleOutput

On exit: if

i \neq j

, then

r0 [kmax \times (j - 1) + i - 1]

contains an estimate of the

(i, j)

th element of the residual cross-correlation matrix at lag zero,

{\hat{R}}_{0}

. When

i = j

r0 [kmax \times (j - 1) + i - 1]

contains the standard deviation of the

i

th residual series. If

fail . code =

NE_G13D_RES or NE_G13D_ZERO_VAR on exit then the first k rows and columns of r0 are set to zero.

14: $r [\dim]$ – doubleOutput

Note: the dimension, dim, of the array r must be at least

kmax \times kmax \times m

Where

R (l, i, j)

appears in this document, it refers to the array element

r [(j - 1) \times kmax \times kmax + (i - 1) \times kmax + l - 1]

On exit:

R (l, i, j)

is an estimate of the

(i, j)

th element of the residual cross-correlation matrix at lag

l

, for

i = 1, 2, \dots, k

j = 1, 2, \dots, k

and

l = 1, 2, \dots, m

. If

fail . code =

NE_G13D_RES or NE_G13D_ZERO_VAR on exit then all elements of r are set to zero.

15: $rcm [pdrcm \times m \times k \times k]$ – doubleOutput

On exit: the estimated standard errors and correlations of the elements in the array r. The correlation between

R (l, i, j)

and

R (l_{2}, i_{2}, j_{2})

is returned as

rcm [pdrcm \times t + s]

where

s = (l - 1) \times k \times k + (j - 1) \times k + i

and

t = (l_{2} - 1) \times k \times k + (j_{2} - 1) \times k + i_{2}

except that if

s = t

, then

rcm [pdrcm \times t + s]

contains the standard error of

R (l, i, j)

. If on exit,

fail . code =

NE_G13D_DIAG or NE_G13D_FACT, then all off-diagonal elements of rcm are set to zero and all diagonal elements are set to

1 / \sqrt{n}

16: $pdrcm$ – IntegerInput

On entry: the first dimension of the array rcm.

Constraint:

pdrcm \geq m \times k \times k

17: $chi$ – double *Output

On exit: the value of the modified portmanteau statistic,

Q_{(m)}^{*}

. If

fail . code =

NE_G13D_RES or NE_G13D_ZERO_VAR on exit then chi is returned as zero.

18: $idf$ – Integer *Output

On exit: the number of degrees of freedom of chi.

19: $siglev$ – double *Output

On exit: the significance level of chi based on idf degrees of freedom. If

fail . code =

NE_G13D_RES or NE_G13D_ZERO_VAR on exit, siglev is returned as one.

20: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_G13D_AR: On entry, the AR parameter estimates are outside the stationarity region.
NE_G13D_ARMA: On entry, $ip = 0$ and $iq = 0$ .
NE_G13D_DIAG: The matrix rcm could not be computed because one of its diagonal elements was found to be non-positive.
NE_G13D_FACT: On entry, the AR operator has a factor in common with the MA operator.
NE_G13D_ITER: Excessive iterations needed to find zeros of determinental polynomials.
NE_G13D_MA: On entry, the MA parameter matrices are outside the invertibility region.
NE_G13D_RES: On entry, at least two of the residual series are identical.
NE_G13D_ZERO_VAR: On entry, at least one of the residual series in the array v has near-zero variance.
NE_INT: On entry, $ip = 〈value〉$ .
Constraint: $ip \geq 0$ .

On entry, $iq = 〈value〉$ .
Constraint: $iq \geq 0$ .

On entry, $k = 〈value〉$ .
Constraint: $k \geq 1$ .
NE_INT_2: On entry, $kmax = 〈value〉$ and $k = 〈value〉$ .
Constraint: $kmax \geq k$ .

On entry, $m = 〈value〉$ and $n = 〈value〉$ .
Constraint: $m < n$ .
NE_INT_3: On entry, $m = 〈value〉$ , $ip = 〈value〉$ and $iq = 〈value〉$ .
Constraint: $m > ip + iq$ .

On entry, $pdrcm = 〈value〉$ , $m = 〈value〉$ and $k = 〈value〉$ .
Constraint: $pdrcm \geq m \times k \times k$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_CLOSE_FILE: Cannot close file $〈value〉$ .
NE_NOT_POS_DEF: On entry, the covariance matrix qq is not positive definite.
NE_NOT_WRITE_FILE: Cannot open file $〈value〉$ for writing.

7

Accuracy

The computations are believed to be stable.

8

Parallelism and Performance

nag_tsa_varma_diagnostic (g13dsc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_tsa_varma_diagnostic (g13dsc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

9.1

Timing

The time taken by nag_tsa_varma_diagnostic (g13dsc) depends upon the number of residual cross-correlation matrices to be computed,

m

, and the number of time series,

k

9.2

Choice of $m$

The number of residual cross-correlation matrices to be computed,

m

, should be chosen to ensure that when the ARMA model (1) is written as either an infinite order autoregressive process, i.e.,

W_{t} - μ = \sum_{j = 1}^{\infty} π_{j} (W_{t - j} - μ) + ε_{t}

or as an infinite order moving average process, i.e.,

W_{t} - μ = \sum_{j = 1}^{\infty} ψ_{j} ε_{t - j} + ε_{t}

then the two sequences of

k

k

matrices

\{π_{1}, π_{2}, \dots\}

and

\{ψ_{1}, ψ_{2}, \dots\}

are such that

π_{j}

and

ψ_{j}

are approximately zero for

j > m

. An overestimate of

m

is therefore preferable to an under-estimate of

m

. In many instances the choice

m = 10

will suffice. In practice, to be on the safe side, you should try setting

m = 20

9.3

Checking a ‘White Noise’ Model

If you have fitted the ‘white noise’ model

W_{t} - μ = ε_{t}

then nag_tsa_varma_diagnostic (g13dsc) should be entered with

p = 1

q = 0

, and the first

k^{2}

elements of par and parhld set to zero and Nag_TRUE respectively.

9.4

Approximate Standard Errors

When

fail . code =

NE_G13D_DIAG or NE_G13D_FACT all the standard errors in rcm are set to

1 / \sqrt{n}

. This is the asymptotic standard error of

{\hat{r}}_{i j} (l)

when all the autoregressive and moving average parameters are assumed to be known rather than estimated.

9.5

Alternative Tests

{\hat{R}}_{0}

is useful in testing for instantaneous causality. If you wish to carry out a likelihood ratio test then the covariance matrix at lag zero

({\hat{C}}_{0})

can be used. It can be recovered from

{\hat{R}}_{0}

by setting

\begin{array}{l} {\hat{C}}_{0} (i, j) & = {\hat{R}}_{0} (i, j) \times {\hat{R}}_{0} (i, i) \times {\hat{R}}_{0} (j, j), & for ​ i \neq j \\ = {\hat{R}}_{0} (i, j) \times {\hat{R}}_{0} (i, j), & for ​ i = j \end{array}

10

Example

This example fits a bivariate AR(1) model to two series each of length

48

μ

has been estimated but

ϕ_{1} (2, 1)

has been constrained to be zero. Ten residual cross-correlation matrices are to be computed.

NAG Library Function Document

nag_tsa_varma_diagnostic (g13dsc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Timing

9.2 Choice of m

9.3 Checking a ‘White Noise’ Model

9.4 Approximate Standard Errors

9.5 Alternative Tests

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

9.1

Timing

9.2

Choice of $m$

9.3

Checking a ‘White Noise’ Model

9.4

Approximate Standard Errors

9.5

Alternative Tests

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results